The absolute Euler product representation of the absolute zeta function for a torsion free Noetherian $\mathbb{F}_1$-scheme

TL;DR

This paper proves that the absolute zeta function of certain schemes over $ extbf{Z}$ can be expressed as an infinite product, called the absolute Euler product, with factors derived from the scheme's counting function over $ extbf{F}_1$.

Contribution

It establishes the absolute Euler product structure for the absolute zeta function of torsion free Noetherian $ extbf{F}_1$-schemes, confirming Kurokawa's suggestion.

Findings

Proves the absolute zeta function has an absolute Euler product structure.

Shows each factor is derived from the counting function of the $ extbf{F}_1$-scheme.

Provides a formulation and proof for Kurokawa's conjecture.

Abstract

The absolute zeta function for a scheme $X$ of finite type over $\mathbb{Z}$ satisfying a certain condition is defined as the limit as $p\to 1$ of the congruent zeta function for $X\otimes\mathbb{F}_p$. In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over $\mathbb{Z}$ should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian $\mathbb{F}_1$-scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the $\mathbb{F}_1$-scheme.